The choice is simple, actually. Do you want to run a blog, or administer a website?
If you just want to run a blog, where you write posts and such, then use WordPress.com.
If you want to run a website, then use the free WordPress software from wordpress.org.
What's the difference, you ask? Well, when you run your own website, you have much more freedom, can use any theme, any plugins you like, and can customize it all you want. But when/if it breaks, it's all on you. You pay for the hosting, you administer the software, it's your site, 100%.
With WordPress.com, you don't have as much control, but you don't have to deal with any of the backend either. You can write posts and such without worrying about it. You can pay a few bucks for your own domain name if you want, and pay a little more to customize your theme via CSS only (surprisingly easy once you get the hang of it). But if you find a theme you like elsewhere that's not on WordPress.com, then you can't use it. If you find a plugin that you think might be useful, and it's not on WordPress.com, then you're out of luck. You give up customizability, but you also don't have to be responsible for the actual site's operation, you don't have to deal with hosting fees and webhost support, etc.
I'd suggest starting a free blog on WordPress.com first. If later you think it's worth the trouble to switch to your own hosting and WordPress.org, then you can still export all your content from one to the other, no problem.